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14
INTENSIONAL PROPERTIES OF POLYGRAPHS
"... Abstract – We present Albert Burroni’s polygraphs as a computational model, showing how these objects can be seen as functional programs. First, we prove that the model is Turing complete. Then, we use a notion of termination proof introduced by the second author to characterize polygraphs that comp ..."
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Abstract – We present Albert Burroni’s polygraphs as a computational model, showing how these objects can be seen as functional programs. First, we prove that the model is Turing complete. Then, we use a notion of termination proof introduced by the second author to characterize polygraphs that compute in polynomial time and, going further, polynomial time functions. 1
Diagram rewriting and operads
, 2009
"... We introduce an explicit diagrammatic syntax for PROs and PROPs, which are used in the theory of operads. By means of diagram rewriting, we obtain presentations of PROs by generators and relations, and in some cases, we even get convergent rewrite systems. This diagrammatic syntax is useful for prac ..."
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We introduce an explicit diagrammatic syntax for PROs and PROPs, which are used in the theory of operads. By means of diagram rewriting, we obtain presentations of PROs by generators and relations, and in some cases, we even get convergent rewrite systems. This diagrammatic syntax is useful for practical computations, but also for theoretical results. Moreover, rewriting is strongly related to homotopy theory. For instance, it can be used to compute homological invariants of algebraic structures, or to prove coherence results.
Polygraphic resolutions and homology of monoids
, 2007
"... We prove that for any monoid M, the homology defined by the second author by means of polygraphic resolutions coincides with the homology classically defined by means of resolutions by free ZMmodules. 1 ..."
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We prove that for any monoid M, the homology defined by the second author by means of polygraphic resolutions coincides with the homology classically defined by means of resolutions by free ZMmodules. 1
Computing Critical Pairs in Polygraphs
 In Workshop on Computer Algebra Methods and Commutativity of Algebraic Diagrams (CAMCAD
, 2009
"... Polygraphs generalize to 2categories the usual notion of equational theory, by describing them as quotients, modulo equations, of freely generated 2categories on a given set of generators. In order to work with morphisms modulo the equations, it is often convenient to orient the equations into a c ..."
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Polygraphs generalize to 2categories the usual notion of equational theory, by describing them as quotients, modulo equations, of freely generated 2categories on a given set of generators. In order to work with morphisms modulo the equations, it is often convenient to orient the equations into a confluent rewriting system. In the case of a terminating system, confluence can be checked by showing that critical pairs are joinable. However, the computation of the critical pairs is more complicated for polygraphs than for term rewriting systems: in particular, two left members of a rule don’t necessarily have a finite number of unifiers. We advocate here that a more general notion of rewriting system should be considered instead, and introduce an operad of compact contexts in a 2category, in which two rules have a finite number of unifiers. A concrete representation of contexts is proposed, as well as an unification algorithm for these.
TOWARDS 3DIMENSIONAL REWRITING THEORY
"... Abstract. String rewriting systems have proved very useful to study monoids. In good cases, they give finite presentations of monoids, allowing computations on those and their manipulation by a computer. Even better, when the presentation is confluent and terminating, they provide one with a notion ..."
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Abstract. String rewriting systems have proved very useful to study monoids. In good cases, they give finite presentations of monoids, allowing computations on those and their manipulation by a computer. Even better, when the presentation is confluent and terminating, they provide one with a notion of canonical representative of the elements of the presented monoid. Polygraphs are a higherdimensional generalization of this notion of presentation, from the setting of monoids to the much more general setting of ncategories. One of the main purposes of this article is to give a progressive introduction to the notion of higherdimensional rewriting system provided by polygraphs, and describe its links with classical rewriting theory, string and term rewriting systems in particular. After introducing the general setting, we will be interested in proving local confluence for polygraphs presenting 2categories and introduce a framework in which a finite 3dimensional rewriting system admits a finite number of critical pairs. Recent developments in category theory have established higherdimensional categories as a fundamental theoretical setting in order to study situations arising in various areas of mathematics, physics and computer science. A nice survey of these can be found in [2],
Team Pareo Formal Islands: Foundations and Applications
"... c t i v it y e p o r t 2008 Table of contents ..."
PROs and diagram rewriting ∗
, 2010
"... We give a survey of a diagrammatic syntax for PROs and PROPs, which are related to the theory of bialgebras. Using diagram rewriting, we obtain presentations of PROs by generators and relations, and in some cases, we even get convergent rewrite systems. Except for Sections 4 and 7, most of the mater ..."
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We give a survey of a diagrammatic syntax for PROs and PROPs, which are related to the theory of bialgebras. Using diagram rewriting, we obtain presentations of PROs by generators and relations, and in some cases, we even get convergent rewrite systems. Except for Sections 4 and 7, most of the material presented here comes from [La03], which was inspired by [Bu93].
TERMINATION ORDERS FOR 3POLYGRAPHS
, 2006
"... aux 3polygraphes, ainsi qu’une application. Abstract: This note presents the first known class of termination orders for 3polygraphs, together with an application. Polygraphs are cellular presentations of higherdimensional categories introduced in [Burroni 1993]. They have been proved to generali ..."
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aux 3polygraphes, ainsi qu’une application. Abstract: This note presents the first known class of termination orders for 3polygraphs, together with an application. Polygraphs are cellular presentations of higherdimensional categories introduced in [Burroni 1993]. They have been proved to generalize term rewriting systems but they lack some tools widely used in the field. This note presents a result developped in [Guiraud 2004] which fills this gap for some 3dimensional polygraphs: it introduces a method to craft termination orders, one of the most useful ways to prove that computations specified by a formal system always end after a finite number of transformations. 1 Notions about 3polygraphs The formal definition of polygraphs can be found in [Burroni 1993]. Here, we restrict ourselves to the case of a 2polygraph with one 0cell and one 1cell: this is a graph Σ over the set of natural numbers. Elements of Σ are called 2dimensional cells or circuits. Two 2cells are parallel when thay have the same source and the same target. A 2dimensional cell ϕ: m → n is graphically pictured as a circuit with m inputs and n outputs: m ϕ n Given such a 2polygraph Σ, one builds another 2polygraph 〈Σ〉: its 2cells are all the circuits one can build from the ones in Σ, by either (horizontal) juxtaposition or (vertical) plugging. These two operations are pictured this way: ( ) ⋆0
Institut de Mathématiques de Luminy (UMR 6206 du CNRS)
, 2006
"... We present various results of the last twenty years converging towards a homotopical theory of computation. This new theory is based on two crucial notions: polygraphs (introduced by Albert Burroni) and polygraphic resolutions (introduced by François Métayer). There are two motivations for such a th ..."
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We present various results of the last twenty years converging towards a homotopical theory of computation. This new theory is based on two crucial notions: polygraphs (introduced by Albert Burroni) and polygraphic resolutions (introduced by François Métayer). There are two motivations for such a theory: • providing invariants of computational systems to study those systems and prove properties about them; • finding new methods to make computations in algebraic structures coming from geometry or topology. This means that this theory should be relevant for mathematicians as well as for theoretical computer scientists, since both may find useful tools or concepts for their own domain coming from the other one. ∗ This work has been partly supported by project GEOCAL (Géométrie du Calcul, ACI Nouvelles Interfaces des Mathématiques) and by project INVAL (Invariants algébriques des systèmes informatiques, ANR). † Address: IML 163 avenue de Luminy, Case 907 13288 Marseille CEDEX 9 Francehttp://iml.univmrs.fr/˜lafont/ 1 Here are the main notions and results presented in this paper: 1. A presentation of a monoid M is convergent if it is noetherian and confluent. Such a presentation can be used to solve the word problem for M [KN85a]. The notion of critical peak is crucial here.